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Combinatorica

, Volume 37, Issue 1, pp 77–86 | Cite as

The (2k-1)-connected multigraphs with at most k-1 disjoint cycles

  • Henry A. Kierstead
  • Alexandr V. Kostochka
  • Elyse C. Yeager
Original Paper

Abstract

In 1963, Corradi and Hajnal proved that for all k≥1 and n≥3k, every (simple) graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k—1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the simple graphs G with minimum degree δ(G)≥2k—1 that do not contain k disjoint cycles. We use this result to answer Dirac's question in full.

Mathematics Subject Classification (2000)

05C15 05C35 05C40 

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References

  1. [1]
    K. Corráadi and A. Hajnal: On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 423–439.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    G. Dirac: Some results concerning the structure of graphs, Canad. Math. Bull. 6 (1963), 183–210.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    G. Dirac and P. Erdős: On the maximal number of independent circuits in a graph, Acta Math. Acad. Sci. Hungar. 14 (1963), 79–94.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    H. A. Kierstead, A. V. Kostochka and E. C. Yeager: On the Corradi-Hajnal Theorem and a question of Dirac, submitted.Google Scholar
  5. [5]
    L. Lovasz: On graphs not containing independent circuits, (Hungarian, English summary) Mat. Lapok 16 (1965), 289–299.MathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Henry A. Kierstead
    • 1
  • Alexandr V. Kostochka
    • 2
    • 3
  • Elyse C. Yeager
    • 2
  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.Sobolev Institute of MathematicsNovosibirskRussia

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